Current-Current Lagrangian

The Fermi effective Lagrangian is written

$${\cal L_{F}} = \frac{G_{F}}{\sqrt{2}} {\cal J}_{\sigma}^{\dagger}(x) {\cal J}^{\sigma}(x)$$ (75)

where the total weak current $J_{\sigma}$ can be separated in the sum of the leptonic weak current and the hadronic weak current

$${\cal J}_{\sigma}(x) = \ell_{\sigma}(x) + h_{\sigma}(x)$$ (76)

For the leptonic current it can be written

$$\ell_{\sigma}(x) = \ell_{\sigma}^{(e)}(x) + \ell_{\sigma}^{(\mu)}(x) + \ell_{\sigma}^{(\tau)}(x)$$ (77)

$$= \bar{\psi}_{\nu_{e}}(x) \gamma_{\sigma} (1-\gamma_{5}) \psi_{e} + [e \rightarrow \mu] + [e \rightarrow \tau]$$ (78)

where its structure of the type $(V-A)$ is evident in the factor $\gamma_{\sigma} (1 - \gamma_{5})$

It is also usual to separate the Lagrangian (77) in the form

$${\cal L}_{F} = {\cal L}_{\ell \ell} + {\cal L}_{\ell h} + {\cal L}_{h h}$$ (79)

each part corresponding to a given type of processes. In fact,

$${\cal L}_{\ell \ell} = \frac{G_{F}}{\sqrt{2}} \{\ell^{(e)\... ...\sigma\dagger} \ell_{\sigma}^{(\mu)} + (\mu,\mu) + [\tau]\}$$ (80)

is referred to purely leptonic weak processes. While

$${\cal L}_{\ell h} = \frac{G_{F}}{\sqrt{2}} \{\ell^{(e)\sig... ...gma} + \ell^{(\mu)\sigma\dagger} h_{\sigma} + [\tau] + h.c.\}$$ (81)

is the part corresponding to semileptonic reactions. It is worth noticing that in the case of the lepton $\tau$, having a mass almost equivalent to two protons, there exist the possibility of semihadronic decays. Finally

$${\cal L}_{ h h} = \frac{G_{F}}{\sqrt{2}} \{ h^{\sigma\dagger} h_{\sigma}\}$$ (82)

will describe the purely hadronic processes.